标题： 非线性sigma模型中的Bałaban变分问题 显示英文标题

标题： The Bałaban variational problem in the non-linear sigma model

摘要： 在由块平均化程序规定的约束下，量子场论（QFT）的作用最小化是Ba{\l }万重整化方法的重要组成部分。 这对于目标空间非平凡的QFT，比如规范理论或格点上的非线性sigma模型，尤其有趣。 我们分析了二维$O(4)$非线性sigma模型的这一步骤，并展示了在这种情况中，Ba{\l }万方法的各种要素是如何共同作用的。 首先，通过李群上的变分微积分，推导出了临界点方程。 然后，通过巴拿赫压缩映射定理解决了这个非线性方程。 这个步骤需要通过对随机行走展开的控制来详细掌握格点格林函数及其积分核。 显示英文摘要

摘要： The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Ba{\l}aban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the $O(4)$ non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Ba{\l}aban's approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.